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Wick polynomials in noncommutative probability: a group-theoretical approach

Part of: Selfadjoint operator algebras General nonassociative rings Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  25 August 2021

Kurusch Ebrahimi-Fard ,
Frédéric Patras ,
Nikolas Tapia  and
Lorenzo Zambotti
Kurusch Ebrahimi-Fard
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway e-mail: [email protected]
Frédéric Patras
Affiliation:
Laboratoire J.A. Dieudonné, Université Côte d’Azur, CNR, UMR 7351, Parc Valrose, Nice, France e-mail: [email protected]
Nikolas Tapia*
Affiliation:
Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany and Institute of Mathematics, Technische Universtät Berlin, Straße des 17. Juni 136, 10587 Berlin, Germany
Lorenzo Zambotti
Affiliation:
Laboratoire de Probabilités, Statistiques et Modélisation, Sorbonne Université, Université de Paris, 4 Place Jussieu, 75005 Paris, France e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, Boolean, and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf-algebraic approach to cumulants and Wick products in classical probability theory.

Keywords

Wick polynomialsmonotone cumulantsfree cumulantsBoolean cumulantsformal power seriescombinatorial Hopf algebrashuffle algebragroup actions

MSC classification

Primary: 17A30: Algebras satisfying other identities 46L53: Noncommutative probability and statistics 46L54: Free probability and free operator algebras
Secondary: 16T05: Hopf algebras and their applications 16T10: Bialgebras 16T30: Connections with combinatorics
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 6 , December 2022 , pp. 1673 - 1699
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work was supported by the European Research Council for Informatics and Mathematics through contract ERCIM 2018-10, and the BMS MATH+ EF1-5 project “On robustness of Deep Neural Networks.”

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